One-second coherence for a single electron spin coupled to a multi-qubit nuclear-spin environment

Delft University of Technology

One-second coherence for a single electron spin coupled to a multi-qubit nuclear-spin

environment

Abobeih, M. H.; Cramer, J.; Bakker, M. A.; Kalb, N.; Markham, M.; Twitchen, D. J.; Taminiau, Tim

DOI

10.1038/s41467-018-04916-z

Publication date

2018

Document Version

Final published version

Published in

Nature Communications

Citation (APA)

Abobeih, M. H., Cramer, J., Bakker, M. A., Kalb, N., Markham, M., Twitchen, D. J., & Taminiau, T. (2018).

One-second coherence for a single electron spin coupled to a multi-qubit nuclear-spin environment. Nature

Communications, 9(1), [2552]. https://doi.org/10.1038/s41467-018-04916-z

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One-second coherence for a single electron spin

coupled to a multi-qubit nuclear-spin environment

M.H. Abobeih

1,2

, J. Cramer

1,2

, M.A. Bakker

1,2

, N. Kalb

1,2

, M. Markham

3

, D.J. Twitchen

3

& T.H. Taminiau

1,2

Single electron spins coupled to multiple nuclear spins provide promising multi-qubit

regis-ters for quantum sensing and quantum networks. The obtainable level of control is

deter-mined by how well the electron spin can be selectively coupled to, and decoupled from, the

surrounding nuclear spins. Here we realize a coherence time exceeding a second for a single

nitrogen-vacancy electron spin through decoupling sequences tailored to its microscopic

nuclear-spin environment. First, we use the electron spin to probe the environment, which is

accurately described by seven individual and six pairs of coupled carbon-13 spins. We

develop initialization, control and readout of the carbon-13 pairs in order to directly reveal

their atomic structure. We then exploit this knowledge to store quantum states in the

electron spin for over a second by carefully avoiding unwanted interactions. These results

provide a proof-of-principle for quantum sensing of complex multi-spin systems and an

opportunity for multi-qubit quantum registers with long coherence times.

DOI: 10.1038/s41467-018-04916-z

OPEN

1_{QuTech, Delft University of Technology, PO Box 50462600 GA Delft, The Netherlands.}2_{Kavli Institute of Nanoscience Delft, Delft University of}

Technology, PO Box 50462600 GA Delft, The Netherlands.3_{Element Six Innovation, Fermi Avenue, Harwell Oxford, Didcot, Oxfordshire OX11 0QR, United}

Kingdom. Correspondence and requests for materials should be addressed to T.H.T. (email:T.H.Taminiau@TUDelft.nl)

123456789

C

oupled systems of individual electron and nuclear spins in

solids are a promising platform for quantum information

processing

1–6

and quantum sensing

7–11

. Initial

experi-ments have demonstrated the detection and control of several

nuclear spins surrounding individual defect or donor electron

spins

12–17

. These nuclear spins provide robust qubits that enable

enhanced quantum sensing protocols

7–11

, quantum error

cor-rection

2,3,18

, and multi-qubit nodes for optically connected

quantum networks

19–22

.

The level of control that can be obtained is determined by the

electron spin coherence and therefore by how well the electron

can be decoupled from unwanted interactions with its spin

environment. Electron coherence times up to 0.56 s for a single

electron spin qubit

5

and

∼3 s for ensembles

23–26

have been

demonstrated in isotopically purified samples depleted of nuclear

spins, but in those cases the individual control of multiple

nuclear-spin qubits is forgone.

Here we realize a coherence time exceeding 1 s for a single

electron spin in diamond that is coupled to a complex

environ-ment of multiple nuclear-spin qubits. First, we use the electron

spin as a quantum sensor to probe the microscopic structure of

the surrounding nuclear-spin environment, including interactions

between the nuclear spins. We

find that the spin environment is

accurately described by seven isolated single

13

C spins and six

pairs of coupled

13

C spins (Fig.

1

a). We then develop pulse

sequences to initialize, control and readout the state of the

13

_C–

13

_{C pairs. We use this control to directly characterize the}

coupling strength between the

13

C spins, thus revealing their

atomic structure given by the distance between the two

13

C atoms

and the angle they make with the magnetic

field. Finally, we

exploit this extensive knowledge of the microscopic environment

to realize tailored decoupling sequences that effectively protect

arbitrary quantum states stored in the electron spin for well over

a second. This combination of a long electron spin coherence

time and selective couplings to a system of up to 19 nuclear spins

provides a promising path to multi-qubit registers for quantum

sensing and quantum networks.

Results

System. We use a single nitrogen-vacancy (NV) center (Fig.

1

a)

in a CVD-grown diamond at a temperature of 3.7 K with a

nat-ural 1.1% abundance of

13

C and a negligible nitrogen

con-centration (<5 parts per billion). A static magnetic

field of Bz

≈

403 G is applied along the NV-axis with a permanent magnet

(Methods). The NV electron spin is read out in a single shot with

an average

fidelity of 95% through spin-selective resonant

exci-tation

27

. The electron spin is controlled using microwave pulses

through an on-chip stripline (Methods).

Longitudinal relaxation. We

first address the longitudinal

relaxation (T1) of the NV electron spin, which sets a limit on the

maximum coherence time. At 3.7 Kelvin, spin-lattice relaxation

due to two-phonon Raman and Orbach-type processes are

neg-ligible

28,29

. No cross relaxation to P1 or other NV centers is

expected due to the low nitrogen concentration. The electron spin

can, however, relax due to microwave noise and laser background

introduced by the experimental controls (Fig.

1

). We ensure a

high on/off ratio of the lasers (>100 dB) and use switches to

suppress microwave amplifier noise (see Methods). Figure

1

b

shows the measured electron spin relaxation for all three initial

states. We

fit the average fidelity F to

F

¼ 2=3e

t=T1

þ 1=3

ð1Þ

The obtained decay time T1

is

ð3:6 ± 0:3Þ ´ 10

3

_{s. This value sets a}

lower limit for the spin relaxation time, and is the longest

reported for a single electron spin qubit. Remarkably, the

observed T1

exceeds recent theoretical predictions based on

single-phonon processes by more than an order of

magni-tude

30,31

. To further investigate the origin of the decay, we

pre-pare ms

= 0 and measure the total spin population summed over

all

three

states.

The

total

population

decays

on

a

similar

timescale

( 3:6 ´ 10

3

_s),

_indicating

_that

_the

decay is caused by a reduction of the measurement contrast,

possibly due to drifts in the optical setup (see Methods),

rather than by spin relaxation. This suggests that the

spin-relaxation time significantly exceeds the measured T1

value.

Nevertheless, the long T1

observed here already indicates that

longitudinal relaxation is no longer a limiting factor for NV

center coherence.

Quantum sensing of the microscopic spin environment. To

study the electron spin coherence, we

first use the electron spin as

a quantum sensor to probe its nuclear-spin environment through

dynamical decoupling spectroscopy

12–14

. The electron spin is

prepared in a superposition

jxi ¼ ðjm

_s

¼ 0i þ jm

_s

¼ 1iÞ=

p

ffiffiffi

2

and a dynamical decoupling sequence of N

π-pulses of the form

ðτ  π  τÞ

N

is applied. The remaining electron coherence is

then measured as a function of the time between the pulses 2τ.

Loss of electron coherence indicates an interaction with the

nuclear-spin environment.

b

a

13_C–13_{C pair} Isolated 13C NV center m_s = 0 m_s = ±1 m_s = 0 m_s = –1 m_s = +1 MW Laser 637 nm Laser 515 nm 12 C 14_N Electron

Total evolution time t (s)

State fidelity 1.0 0.8 0.6 0.4 10–2 ₁₀–1 ₁₀0 ₁₀1 ₁₀2 ₁₀3 ₁₀4 ₁₀5

Fig. 1 Experimental system and_T1measurements.a We study a single

nitrogen-vacancy (NV) center in diamond surrounded by a bath of13_C

nuclear spins (1.1% abundance). In this work, we show that the microscopic nuclear-spin environment is accurately described by 7 isolated13C spins, 6 pairs of coupled13_{C spins and a background bath of}13_{C spins (not}

depicted).b Longitudinal relaxation of the NV electron spin. The spin is prepared inm_s¼ 0; 1, or + 1 and the fidelity with the initial state is measured after time_{t. The inset shows the microwave (MW) and laser} controls for the NV spin and charge states, as well as the pathways for spin relaxation induced by potential background noise from these controls. All error bars are one statistical s.d.

The results in Fig.

2

a for N

= 32 pulses reveal a rich structure

consisting of both sharp and broader dips in the electron

coherence. The sharp dips (Fig.

2

b) have been identified

previously as resonances due to the electron spin undergoing

an entangling operation with individual isolated

13

C spins in the

environment

12–14

. For this NV center, the observed signal is well

explained by seven individual

13

C spins and a background bath of

randomly generated

13

C spins (Fig.

2

b). To verify this explanation

we perform direct Ramsey spectroscopy on all seven spins

(Supplementary Fig.

1

)

3

. For the electron spin in m

s

¼ ± 1, each

spin yields a single unique precession frequency due to the

hyperfine coupling, indicating that all seven spins are distinct and

do not couple strongly to other

13

C spins in the vicinity

(Supplementary Fig.

1

).

The electron can be efficiently decoupled from the interactions

with such isolated

13

C spins by setting

τ ¼ m 

2π_ω

L

, with m a

positive integer and

ωL

the

13

C Larmor frequency for ms

= 0

33

. In

practice, however, this condition might not be exactly and

simultaneously met for all spins due to: the limited timing

resolution of

τ (here 1 ns), measurement uncertainty in the value

ωL, and differences between the ms

= 0 frequencies for different

13

_{C spins, for example caused by different effective g-tensors}

a

b

c

1 2 3 4 5 1 6 N/2  2    1.0 1.0 0.5 0.0 1.0 0.5 0.0 0.8 0.6 0.4 0.2 0.0 20 10 58 60 62 64  (µs) 66 C1 C2 C3 C4 Total Pair 1 C5 C6 C7 Bath 11 12 13 14 15 40 60 80 100 120 140 160 180 200 220 240 260 280 State fidelity State fidelity State fidelity

Fig. 2 Quantum sensing of the microscopic spin environment. a Dynamical decoupling spectroscopy13revealing a rich nuclear-spin environment consisting of individual13C spins, as well as pairs of coupled13C spins. The electron spin is prepared in a superposition,jxi ¼ ðjm_s¼ 0i þ jm_s¼ 1iÞ=pffiffiffi2and a decoupling sequence ofN = 32 π-pulses separated by 2τ is applied. Loss of coherence indicates the interaction of the electron spin with nuclear spins in the environment. Blue: data. Purple line: theory (see Methods). The shaded areas mark the signals due to six13C–13C pairs labeled 1–6. b Zoom-in showing sharp signals due to coupling to isolated individual13_{C spins}12–14_{. The total signal is well described by seven}13_{C spins (see Supplementary Table2for}

hyperfine parameters) and a bath of 200 randomly generated spins with hyperfine couplings below 10 kHz. c Zoom-in showing a broad signal due to13C–

13_{C pair 1}16,32_{. Blue: data. The solid orange line is the theoretical signal just due to pair 1, while the purple line includes the seven individual}13_{C spins and the} 13_{C spin bath as well}

under a slightly misaligned magnetic

field (here <0.35°,

Supplementary Note

3

)

3,33–35

. We numerically simulate these

deviations from the ideal condition and

find that, for our range of

parameters, the effect on the electron coherence is small and can

be neglected (Supplementary Fig.

2

).

We associate the broader dips in Figs.

2

a and

2

c to pairs of

strongly coupled

13

C spins. Such

13

C–

13

C pairs were treated

theoretically

32,36

and the signal due to a single pair of

nearest-neighbor

13

C spins with particularly strong couplings to a NV

center has been detected

16

. In this work, we exploit improved

coherence times to detect up to six pairs, including previously

undetected non-nearest-neighbor pairs. We then develop pulse

sequences to polarize and coherently control these pairs to be able

to directly reveal their atomic structure through spectroscopy.

Direct spectroscopy of nuclear-spin pairs. The evolution of

13

C–

13

_{C pairs can be understood from an approximate pseudo-spin}

model in the subspace spanned by |"#i ¼ j *i and |#"i ¼ j +i,

following Zhao et al.

32

(Supplementary Notes

1

and

2

). The

pseudo-spin Hamiltonian depends on the electron spin state. For

ms

= 0 we have:

^H

₀

¼ X^S

x

;

ð2Þ

and for m

_s

¼ 1:

^H

₁

¼ X^S

x

þ Z^S

z

;

ð3Þ

where ^S

_x

and ^S

_z

are the spin−

1

2

operators. X is the dipolar

cou-pling between the

13

C spins and Z is due to the hyperfine field

gradient (Supplementary Note 2)

32

. The evolution of the

13

C–

13

C

pair during a decoupling sequence will thus in general depend on

the initial electron spin state, causing a loss of electron coherence.

We now show that this conditional evolution enables direct

spectroscopy of the

13

C–

13

C dipolar interaction X. Consider two

limiting cases: X>>Z and Z>>X, which cover the pairs observed

in this work. In both cases, loss of the electron coherence is

expected for the resonance condition

τ ¼ τ

_k

¼ ð2k  1Þ

_2ωπ

r

, with

k

a

positive

integer

and

resonance

frequency

ω

r

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

X

2

þ ðZ=2Þ

2

q

13,32,37

_{. For X>>Z the net evolution at}

resonance is a rotation around the z-axis with the rotation

direction conditional on the initial electron state (mathematically

analogous to the case of a single-

13

C spin in a strong magnetic

field

13,38

_{). For Z>>X the net evolution is a conditional rotation}

around the x-axis (analogous to the nitrogen nuclear spin

subjected to a driving

field

37

). These conditional rotations provide

the controlled gate operations required to initialize, coherently

control and directly probe the pseudo-spin states.

The measurement sequences for the two cases are shown in

Fig.

3

a. First, a dynamical decoupling sequence is performed that

correlates the electron state with the pseudo-spin state. Reading

out the electron spin in a single shot then performs a projective

measurement that prepares the pseudo-spin into a polarized state.

For X>>Z the pseudo-spin is measured along its z-axis and thus

prepared in |*i. For Z>>X the measurement is along the x-axis

and the spin is prepared in

ðj *i þ j +iÞ=

p

ffiffiffi

2

. Second, we let the

pseudo-spin evolve freely with the electron spin in one of its

eigenstates (ms

= 0 or m

s

¼ 1) so that we directly probe the

precession frequencies

ω

0

¼ X (for ms

= 0) or ω

1

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

X

2

þ Z

2

p

(for m

_s

¼ 1). For Z>>X, an extra complication is that the initial

state

ðj *i þ j +iÞ=

p

ffiffiffi

2

is an eigenstate of ^

H

₀

. To access

ω

₀

¼ X,

we prepare

ðj *i þ ij +iÞ=

p

ffiffiffi

2

− a superposition of ^H

0

eigenstates

—by first letting the system evolve under ^H

1

for a time

π=ð2ω

1

Þ.

Finally, the state of the pseudo-spin is readout through a second

measurement sequence.

We

find six distinct sets of frequencies (Fig.

3

b), indicating that

six different

13

C–

13

C pairs are detected. The measurements for

ms

= 0 directly yield the coupling strengths X and therefore the

atomic structure of the pairs (Fig.

4

a). We observe a variety of

coupling strengths corresponding to nearest-neighbor pairs

(X=2π ¼ 2082:7ð7Þ Hz, theoretical value 2061 Hz), as well as

pairs separated by several bond lengths (e.g., X=2π ¼ 133:8ð1Þ

Hz, theoretical value 133.4 Hz). The observed number of pairs is

consistent with the

13

C concentration of the sample

(Supple-mentary Fig.

4

). Note that for pair 4, we have X>>Z, so the

resonance condition is mainly governed by the coupling strength

X. This makes it likely that additional pairs with the same dipolar

coupling X—but smaller Z-values—contribute to the observed

signal at

τ ¼ 120 µs. Nevertheless, the environment can be

described accurately by the six identified pairs, which we verify by

comparing the measured dynamical decoupling curves for

different values of N to the calculated signal based on the

extracted couplings (Fig.

4

b).

Electron spin coherence time. Next, we exploit the obtained

microscopic picture of the nuclear spin environment to

investi-gate the electron spin coherence under dynamical decoupling. To

extract the loss of coherence due to the remainder of the

dynamics of the environment, i.e., excluding the identified signals

from the

13

C spins and pairs, we

fit the results to:

F

¼

1

2

þ A  MðtÞ  e

ðt=TÞn

;

ð4Þ

in which M(t) accounts for the signal due to the coupling to the

13

_C–

13

_{C pairs (Fig.}

₄

_{b, Methods section). A, T, and n are}

_fit

parameters that account for the decay of the envelope due to the

rest of the dynamics of the environment and pulse errors. As

before, effects of interactions with individual

13

C spins are

avoi-ded by setting

τ ¼ m 

2π_ω

L

. An additional challenge is that at high

numbers of pulses the electron spin becomes sensitive even to

small effects, such as spurious harmonics due to

finite MW pulse

durations

39,40

and non-secular Hamiltonian terms

41

, which cause

loss of coherence over narrow ranges of

τ (<10 ns). Here we avoid

such effects by scanning a range of

∼20 ns around the target value

to determine the optimum value of

τ.

Figure

5

a shows the electron coherence for sequences from N

= 4 to 10,240 pulses. The coherence times T, extracted from the

envelopes, reveal that the electron coherence can be greatly

extended by increasing the number of pulses N. The maximum

coherence time is T

= 1.58(7) s for N = 10,240 (Fig.

5

b). We

determine the scaling of T with N by

fitting to T

_N¼4

 ðN=4Þ

η

,

with TN=4

the coherence time for N

= 4

23,42–45

which gives

η =

0.799(2). No saturation of the coherence time T is observed yet,

so that longer coherence times are expected to be possible. In our

experiments, however, pulse errors become significant at larger N,

causing a decrease in the amplitude A (Supplementary Fig.

7

).

Protecting arbitrary quantum states. Finally, we demonstrate

that arbitrary quantum states can be stored in the electron spin

for well over a second by using decoupling sequences that are

tailored to the specific microscopic spin environment (Fig.

5

c).

For a given storage time, we select

τ and N to maximize the

obtained

fidelity by avoiding interactions with the characterized

13

_{C spins and}

13

_C–

13

_{C pairs. To assess the ability to protect}

arbitrary quantum states, we average the storage

fidelity over the

six cardinal states and do not re-normalize the results. The results

show that quantum states are protected with a

fidelity above the

2/3 limit of a classical memory for at least 0.995 seconds (using N

= 10,240 pulses) and up to 1.46 s from interpolation of the

results. These are the longest coherence times reported for single

solid-state electron spin qubits

5

, despite the presence of a dense

nuclear spin environment that provides multiple qubits.

Discussion

These results provide opportunities for quantum sensing and

quantum information processing, and are applicable to a wide

variety of solid-state spin systems

4,5,17,46–56

. First, these

experi-ments are a proof-of-principle for resolving the microscopic

structure of multi-spin systems, including the interactions

between spins

32

. The developed methods might be applied to

detect and control spin interactions in samples external

to

the

host

material

10,57–59

.

Second,

the

combination

of long coherence times and selective control in an

electron-nuclear system containing up to twenty spins enables improved

multi-qubit

quantum

registers

for

quantum

networks.

The electron spin coherence now exceeds the time needed to

entangle

remote

NV

centers

through

a

photonic

link,

making deterministic entanglement delivery possible

60

.

More-over, the realized control over multiple

13

C–

13

C pairs provides

promising multi-qubit quantum memories with long coherence

times, as the pseudo-spin naturally forms a

decoherence-protected subspace

61

.

Methods

Setup. The experiments are performed at 3.7 K (Montana Cryostation) with a magneticfield of ∼403 G applied along the NV-axis by a permanent magnet. We realize long relaxation (T1>1 h) and coherence times (>1 s) in combination with

fast spin operations (Rabi frequency of 14 MHz) and readout/initialization (∼10 μs), by minimizing noise and background from the microwave (MW) and optical controls. Amplifier (AR 25S1G6) noise is suppressed by a fast microwave switch (TriQuint TGS2355-SM) with a suppression ratio of 40 dB. Video leakage noise generated by the switch isfiltered with a high pass filter. We use Hermite pulse envelopes62,63to obtain effective MW pulses without initialization of the intrinsic

14_{N nuclear spin. To mitigate pulse errors we alternate the phases of the pulses}

following the XY8 scheme64_{. Laser pulses are generated by direct current}

mod-ulation (515 nm laser, Cobolt MLD - for charge state control) or by acoustic optical modulators (637 nm Toptica DL Pro and New Focus TLB-6704-P for spin pumping and single-shot readout27). The direct current modulation yields an on/ off ratio of >135 dB. By placing two modulators in series (Gooch and Housego Fibre Q) an on/off ratio of >100 dB is obtained for the 637 nm lasers. The laser frequencies are stabilized to within 2 MHz using a wavemeter (HF-ANGSTROM WS/U-10U). Possible explanations for the observed decay in Fig.1b are frequency drifts of this wavemeter or spatial drifts of the laser focus over 1-h timescales.

Sample. We use a naturally occurring NV center in high-purity type IIa homo-epitaxially chemical-vapor-deposition (CVD) grown diamond with a 1.1% natural abundance of13C and a〈111〉 crystal orientation (Element Six). The NV center studied here has been selected for the absence of very-close-by strongly coupled

13_{C spins (>500 kHz hyperfine coupling), but not on any other properties of the}

nuclear spin environment. To enhance the collection efficiency a solid-immersion

X X z z

b

0.6 0.5 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.7 0.7 5 10 15 20 5 0 1 2 3 50 100 150 200 10 15 20 40 60 80 0.3 0.4 0 0.7 0.6 0.5 0.4 10 20 30

a

y ±x x x y ±x y ± z x x y ±z 0 State fidelity Pair 6 Pair 4 Pair 2 Pair 6 Pair 4 Pair 2 X t t m m 0 0 X X

Initialize along z Measure along z

Initialize along x Measure along x

Z >> X

X >> Z

Free evolution time t (ms) Free evolution time t (ms) 0 0.247(5) kHz 6.587(7) kHz 2.0827(7) kHz 0.1338(1) kHz 1.831(3) kHz 2.0843(2) kHz

c

Fig. 3 Direct spectroscopy of nuclear-spin pairs. a Measurement sequences for Ramsey spectroscopy of13C–13C pairs, forZ>>X and for X>>Z. The controlled ±x (±z) gates are controlled ±π/2 rotations around x (z) with the sign controlled by the electron. The initial states are ρ₀¼ j0ih0j and the mixed stateρ_m.b, c Nuclear spin Ramsey measurements and obtained precession frequencies for pairs 2, 4, and 6. The electron spin state during the free evolution timet is set to ms= 0 (b) or ms¼ 1 (c) and an artificial detuning is applied. Each pair yields a unique set of frequencies, confirming that the

pairs are distinct. For pair 2 an additional beating is observed (frequency of 23(3) Hz), indicating a small coupling to one (or more) additional spins. See Supplementary Fig.3for the other three pairs and Supplementary Table4forfit results. All error bars are one statistical s.d.

b

N = 32 N = 16

a

z = 0 0 0 0 0 z = 0 0 0 Pairs 1,2 _1.0 1.0 0 0 50 50 100 100 150 150 200 200 250 250 Pair 1 Pair 2 Pair 3 Pair 4 Pair 5 Pair 6 Total 300  (µs)  (µs) 300 350 350 400 400 450 500 0.8 0.8 0.6 0.6 State fidelity State fidelity Pair 3 Pair 4 Pair 5 Pair 6 a0 z = 0 z = 0 z = 0 ½ ¾ ½ ¾ ¼ ¼ ½ ½ ½ 1¼ ¼ ¼ ¼ ¼

Fig. 4 Atomic structure and decoupling signal for the six nuclear-spin pairs. a Structure of the six13_C–13_{C pairs within the diamond unit cell (up to}

symmetries and equivalent orientations). Thez-values give the height in fractions of the diamond lattice constant a0. The magneticfield is oriented along

the <111> direction, i.e., along the axis of pair 4. For pair 3 there is an additional possible structure that yields a similarX, Supplementary Table3.b The calculated signal for the six individual13_C–13_{C pairs accurately describes the measured decoupling signal for different number of pulsesN. Data are taken}

forτ ¼ m 2π_ωLto avoid coupling to single-13C spins. See Supplementary Fig.5for other values ofN

b

a

c

N 1.0 100 103 103 ₁₀4 ₁₀0 ₁₀1 ₁₀2 ₁₀3 ₁₀4 102 102 101 101 101 ₁₀2 ₁₀3 0.9 0.8 0.7 0.6 0.5 0.4 Normalized signal

Total evolution time t (ms)

1.0

Classical limit

Average state fidelity

0.9 0.8 0.7 0.6 0.5 Coherence time T (ms) N 4 8 16 32 64 128 256 512 1024 2048 3072 6144 10,240 32 64 128 512 1024 2048 3072 4096 6144 10,240 20,480

Total evolution time t (ms) N

Fig. 5 Protecting quantum states with tailored decoupling sequences. a Normalized signal under dynamical decoupling with the number of pulses varying fromN = 4 to N = 10,240. The electron is initialized and readout along x. The thin lines are fits to equation (4), which takes into account the six identified

13_C–13

C pairs. We use the extracted amplitudesA to re-normalize the signal. Thick lines are the extracted envelops 0:5 þ 0:5  e ðt=TÞnwithT and n obtained from thefits. See Supplementary Fig.6for the obtained valuesn. b Scaling of the obtained coherence time T as function of the number of pulses (error bars are <5%). The solid line is afit to the power function T_N¼4 ðN=4Þη, whereTN=4is the coherence time forN = 4. We find η = 0.799(2). c The

average statefidelity obtained for the six cardinal states (Supplementary Fig.8). Unlike ina, the signal is shown without any renormalization. The number of pulsesN is chosen to maximize the obtained signal at the given total evolution time while avoiding interactions with the13C environment. The solid green line is a_{fit to an exponential decay. The horizontal line at}2

3fidelity marks the classical limit for storing quantum states. The two curves cross at t =

1.46 s demonstrating the protection of arbitrary quantum states well beyond a second. All error bars are one statistical s.d.

lens was fabricated on top of the NV center27,65and a single-layer aluminum-oxide anti-reflection coating was deposited66,67_.

Data analysis. We describe the total signal for the NV electron spin after a decoupling sequence in Fig.2as:

F¼1 2þ A  MbathðtÞ  Y7 i¼1 Mi CðtÞ  Y6 j¼1 MjpairðtÞ  eðt=TÞ n ; ð5Þ where t is the total time. Mbathis the signal due to a randomly generated

back-ground bath of non-interacting spins with hyperfine couplings below 10 kHz. Mi C

are the signals due to the seven individual isolated13_{C spins}13_{. M}j

pairare the signals

due to the six13_C–13_{C pairs and are given by 1=2 þ ReðTrðU}

0U1yÞÞ=4, with U0and

U1the evolution operators of the pseudo-spin pair for the decoupling sequence

conditional on the initial electron state (ms= 0 or ms¼ 1)32. The coherence time

T and exponent n describe the decoherence due to remainder of the dynamics of the spin environment.

Settingτ ¼ m  2π=ωLavoids the resonances due to individual13C spins, so that

equation (5) reduces to: F¼1 2þ A  Y6 j¼1 Mpairj ðtÞ  eðt=TÞ n : ð6Þ

The data in Figs.3and4arefitted to equation (6) and A, T and n are extracted from thesefits.

Data availability. The data that support thefindings of this study are available from the corresponding author upon request.

Received: 4 January 2018 Accepted: 21 May 2018

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Acknowledgements

We thank V. V. Dobrovitski, J. E. Lang, T. S. Monteiro, H. P. Bartling, C. L. Degen, and R. Hanson for valuable discussions, P. Vinke, R. Vermeulen, R. Schouten, and M. Eschen for help with the experimental apparatus, and A. J. Stolk for characterization measure-ments. We acknowledge support from the Netherlands Organization for Scientific Research (NWO) through a Vidi grant.

Author contributions

M.H.A. and T.H.T. devised the experiments. M.H.A., J.C., and T.H.T. constructed the experimental apparatus. M.M. and D.J.T. grew the diamond. M.H.A. performed the experiments with support from M.A.B. and N.K. M.H.A. and T.H.T. analyzed the data with help of all authors. T.H.T. supervised the project.

Additional information

Supplementary Informationaccompanies this paper at https://doi.org/10.1038/s41467-018-04916-z.

Competing interests:The authors declare no competing interests.

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